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How to Use CORDIC

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How to Use CORDIC Topic XC886/888CLM Cordic & MDU Module May 2006 n o Department e n i f n XC886/888CLM May 2006 I Page 1 Agenda XC888/886CLM – Cordic Coprocessor XC886/888CLM – MDU Coprocessor XC886/888CLM May 2006 Page 2 XC886/888CLM – New Peripherals T21 UART1 MultiCan Cordic MDU XC886/888CLM May 2006 Page 3 XC886/888CLM – Cordic Basics Cordic (COordinate Rotation DIgital Computer) – Mathematical co-processor for 16bit trigonometric, hyperbolic and linear functions (e.g. to solve SIN, COS, LOG, EXP, SQRT...) – Hardcoded - Look Up Table based on iterative approximation algorithm (16 iterations, max 41 cycles) 8 SFRs in the mapped area one interrupt vector on a shared node @0x0043 XC886/888CLM May 2006 Page 4 XC886/888CLM – Cordic History CORDIC algorithms belong to the class of shift-add algorithms. (introduced by Jack E. Volder in 1951 for trigonometric functions, generalized and extended with linear and hyperbolic functions by J.S. Walther 1971 ) The idea behind the CORDIC is, to have all mathematic functions in a microcontroller available with most less costs, since shift and add hardware can be easy implemented. Mainly the CORDIC is used to perform a rotation in a plane using a series of specific incremental rotation angles selected so that each is performed by a shift and add operation. There is also a mechanism for computing the magnitude and phase angle of an input vector. One CORDIC iteration involves 2 shifts, 1 table lookup, 3 additions XC886/888CLM May 2006 Page 5 XC886/888CLM – Some Math Basics Some basic functions Taylor series expansion – sine (x), cosine (x) x3 x5 x7 sin x = x − + − ±...; 3! 5! 7! –arctan(x) x3 x5 x7 arctan x = x − + − ±... 3 5 7 Factorial – e.g. 5! = 5*4*3*2*1 = 120 XC886/888CLM May 2006 Page 6 XC886/888CLM – Some Math Basics cont. The Unite Circle Radian π – deg to rad angle []rad = angle [deg ]• 180 e.g. 45 (deg) Æ 45/180*PI = 0.7854 (rad) e.g. 1 (rad) = PI Æ 180 (deg) XC886/888CLM May 2006 Page 7 XC886/888CLM – Cordic Coordinate Systems CORDIC equations m Coordinate Domain of System Convergence (i+1) (i) (i) –i x = x – m di y 2 (i+1) (i) (i) –i y = y + di x 2 1 circular 1.74 radian (i+1) (i) (i) z = z – di e 0 linear 2 -1 hyperbolic 1.11 radian XC886/888CLM May 2006 Page 8 XC886/888CLM – Cordic Iterations Starting with a phase of 45 degrees, the phase of each successive R multiplier is a little over half of the phase of the previous “Binary search" on phase by adding or subtracting successively smaller phases to reach the "target" phase –––––––––––––––––––––––––––––––– i e(i) in degrees e(i) in radians ––––––––––––––––––––––––––––––– 0 45.0 0.785 398 163 1 26.6 0.463 647 609 2 14.0 0.244 978 663 3 7.1 0.124 354 994 4 3.6 0.062 418 810 5 1.8 0.031 239 833 6 0.9 0.015 623 728 7 0.4 0.007 812 341 8 0.2 0.003 906 230 9 0.1 0.001 953 123 –––––––––––––––––––––––––––––––– Example: 30° angle XC886/888CLM 30.0 ≅ 45.0 – 26.6 + 14.0 – 7.1 + 3.6 + 1.8 – 0.9 + 0.4 – 0.2 + 0.1 May 2006 Page 9 = 30.1 XC886/888CLM – Cordic Math Functions CORDIC Rotation Vectoring Linear IN xn,zn,yn xn,yn,zn MUL/DIV/MAC OUT y =x * z + y z =y / x + z K = 1 n+1 n n n n+1 n n n Circular IN zn=angle, xn=K,yn=0 xn,yn, zn=0 SINE/COSINE angle/ OUT xn+1 = cos(angle) zn+1 = arctan(yn/xn ) magnitude 2 + 2 1/2 yn+1 = sin(angle) xn+1 = 1/K*(xn yn ) K = 1.64676 IN z1=angle, x1=K´,y1=0 y1 <x1, z1=0 Hyperbolic K = 0.828 OUT xn+1 = cosh(angle) zn+1 = arctanh(y1/x1) XC886/888CLM 2 - 2 1/2 May 2006 yn+1 = sinh(angle) xn+1 = 1/K´*(x1 y1 ) Page 10 XC886/888CLM – Cordic Data Format S15 Example S15 – S15 or e.g. S5.10 or S10.5. 25 x -25 = -625 – signed integer 0x0019 x 0xFFE7 = 0xFD8F one sign bit | 15 number bits 2.5 x -2.5 = -6.25 – rational number (fractional) 0x0028 x 0xFFD8 = 0xF9C0 one sign | n . (15-n) bit fractional 0.25 x -0.25= 0.0625 decimal point can be freely chosen fraction means S | 21 + 20. 2-1 + 2-2 + 2-3 +.... – negative numbers in Twos Complement e.g. -25dec = 1|111 1111 1110 0110 +1 = 1|111 1111 1110 0111 4Q16 (S4.11) – decimal point is fixed Example S4.11 S| 4.11 2.5 = 0|001‘0.100‘0000‘0000 = 0x1400 XC886/888CLM (-5.625 = 0|010‘1.101‘0000‘0000 = 0x2D00 May 2006 Page 11 twos compl)= 1|101‘0.010‘1111‘1111 + 1 = 0xD300 XC886/888CLM – Cordic Data Format cont. Input: Results: – X, Y can be integer (S15) or rational – X, Y can be integer (S15) or rational number (fraction) in all operating number (fraction) in all operating modes. They must have the same modes. Must be same format as form and in case of fraction the same input format. number of bits for the decimal place. –X in circular vectoring mode can be interpreted as twos complement or as 16-bit unsigned depending on bit X_USIGN X, Y Result Data = CD_Result / MPS MPS = magnitude prescaler (1, 2, 4) – Z for linear functions it is always in – Z for linear functions is signed 4Q16 signed 4Q16 (S4.11) format ( S4.11 ) – Z for circular and hyperbolic functions – Z for circular and hyperbolic it is always handled functions is a scaled normalized as scaled normalized integer. integer XC886/888CLM May 2006 Page 12 input Z = real Z (in rad) * 32768/PI real result Z (in rad) = Z* PI/32768 XC886/888CLM – Cordic Accuracy & Deviation Input inaccuracy: representation of a value is limited by the number of bits The CORDICs accuracy is also limited by the number of iterations. Therefore an absolute deviation can be calculated. This is the difference between the expected and the real value. The normalized deviation refers to the magnitude of the result, i.e. the absolute error for small numbers is big, but if this is put into relationship to the magnitude it is still small. Properly scaled values will produce small errors. The max. error can be calculated and is depending on the chosen mode. Some examples for Linear and Circular mode XC886/888CLM May 2006 Page 13 XC886/888CLM – Cordic Accuracy & Deviation cont Some example calculation in LINEAR mode – y = z * x |z| ≤ 2 Æ z in S1.11 with x = 3000, 6000, ....30000 in S15 and z = 1, 0.1, 0.01, 0.001 in S4.11 // result magnitude in S15 = 32768 // calculate 0*0.01, 3000*0.01, 6000*0.01 ... 30000*0.01 for (x=0;x<32000;x+=3000){ //(x, z=0.01 S4Q16) product = CD_multiply (x, 0x0014); } – here: the deviation depends on how accurate the value z can be coded in S4.11 (Input inaccuracy) Æ max. error: 2-11 = 0,000488281 Z 1 0.1 0.01 0.001 hex 0x0800 0x00CD 0x0014 0x0002 XC886/888CLM Zcoded May 2006 dec 1.000000 0.100098 0.009766 0.000977 Page 14 XC886/888CLM – Cordic Accuracy & Deviation cont X 3000 6000 9000 12000 15000 18000 21000 24000 27000 30000 Z 1 3000 6000 9000 12000 15000 18000 21000 24000 27000 30000 0x0800 0.1 300 600 901 1201 1501 1802 2102 2403 2701 3003 0x00CD 0.01 0x0014 29 58 88 117 146 176 205 235 264 293 0.001 3 6 9 12 15 18 21 24 27 30 0x0002 – Input format of X: integer (S15) or fraction – It is necessary to reserve as many bits as possible after the decimal point to minimize the output truncation error of the result. i.e. 18.5 Æ possible: 0|000 0001 0010.1000 = 0x0128 better: 0|001 0010.1000 0000 = 0x1280 XC886/888CLM May 2006 Page 15 XC886/888CLM – Cordic Scaling the Amplitude Example for circular rotation mode – sine calculation –YFINAL = K*X*sin(Z) K = 1,64676 – For scaling the amplitude A = K*X, set X = A/K note: choose the amplitude as high as possible to increase the resolution of the range of sin(Z) and to decrease the result error note: Amax = 32768 X A YFINAL 1 1.64676 {-1.64676…+1.64676} 100 164.676 {-164.676…+164.676} 12145 19999.9 {-19999.9…+19999.9} 19898 32767.3 {-32767.3…+32676.3} – Angle Z in normalized scaled integer ° 180 = ° XC886/888CLM Min. adjusting angle: 0.00549 May 2006 32768 Page 16 XC886/888CLM – Cordic Example SINE() z0=angle, x0=K,y0=0 How to calculate sine(z) yn+1 = sine(angle) SYSCON0 |= 0x01; // switch into mapped area CD_STATC = 0x00; // clear keep bits - disable CORDIC IRQ CD_CON = 0x0A; // circular rotation mode - auto start CD_CORDYL= 0x00; CD_CORDYH= 0x00; // y0=0 CD_CORDXH= 0x17; // x0hi=K CD_CORDZL= z; CD_CORDZH= (z>>8); // input angle z in integer ~1µs CD_CORDXL= 0xB9; // x0lo=K and start Cordic while(!(CD_STATC & 0x04)); // wait on CORDIC ready printf ("SIN CORDIC = %d%d\n",CD_CORDYH,CD_CORDYL ); // print result SYSCON0 &= 0xFE; // switch to non-mapped area XC886/888CLM May 2006 Page 17 XC886/888CLM – Cordic Example ARCTAN() xn, yn, zn=0 How to calculate an angle zn+1 = arctan(yn/xn ) CD_S15 CORDIC_CalculateAtan (CD_S15 Xval ,CD_S15 Yval) { if(Xval == 0x0000) { g_CORDIC_atan.WORD = 0; // Avoid error at 'divide by 0' } else { ´ SYSCON0 |= 1; CD_STATC = 0; CD_CON =(MPS_1+TWOS_COMPLEMENT+START_AUTO+MODE_VECTORING+FUNC_CIRCULAR); CD_CORDZH = CD_CORDZL = 0; CD_CORDYH = (unsigned char) Yval; CD_CORDYL = (unsigned char) (Yval>>8); CD_CORDXH = (unsigned char) Xval; CD_CORDXL = (unsigned char) (Xval>>8); while(!(CD_STATC & 0x04)); // wait on CORDIC ready g_CORDIC_atan.BYTE.H = CD_CORDZH; ~1µs g_CORDIC_atan.BYTE.L = CD_CORDZL; SYSCON0 &= 0xFE; } XC886/888CLM May 2006 } Page 18 XC886/888CLM – Cordic Example Multiply & Accumulate xn, zn, yn How to do a MAC in S4Q16 (S4.11) yn+1 = xn * zn + yn // mac (multiply & accumulate) -- input format x, y in S15, z in S4.11 // x=4096 = 0x1000, y=4096 =0x1000, z=2 =0x1000 >> yn+1 = 12288 = 0x3000 SYSCON0 |= 0x01; // set RMAP ~1µs CD_CON =(MPS_1+TWOS_COMPLEMENT+START_WITH_ST_SET+MODE_ROTATION+FUNC_LINEAR); CORDIC_LOAD_XYZ_REGISTERS(0x1000,0x1000,0x1000); // (x, y, z) CORDIC_START(); // set start bit while(!(CD_STATC & 0x04)); // Polling until CORDIC is ready – EOCflag g_CORDIC_mac.BYTE.H = CD_CORDYH; g_CORDIC_mac.BYTE.L = CD_CORDYL;
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